A075084 Number of composite numbers c with n <= c <= 2*n.

0, 1, 2, 3, 4, 5, 5, 7, 7, 7, 8, 9, 10, 12, 12, 12, 13, 15, 15, 17, 17, 17, 18, 19, 20, 21, 21, 22, 23, 24, 24, 26, 27, 27, 28, 28, 28, 30, 31, 31, 32, 33, 34, 36, 36, 37, 38, 40, 40, 41, 41, 41, 42, 43, 43, 44, 44, 45, 46, 48, 49, 51, 52, 52, 53, 53, 54, 56, 56, 56, 57, 59, 60

Offset: 1

Author

Amarnath Murthy, Sep 11 2002

Keywords

• easy
• nonn

Comments

The number of composite numbers <= n is n less the number of primes less 1.

The sequence is nondecreasing.

Examples

			a(8) = 7: the composite numbers are 8,9,10,12,14,15 and 16.
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Crossrefs

Related sequences:

Primes (p) and composites (c): A000040, A002808, A000720, A065855.

Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.

Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

Programs

Maple

chi := proc(n) if n <= 3 then 0 else n - numtheory:-pi(n) - 1; fi; end; # A065855
A075084 := proc(n) chi(2*n) - chi(n-1); end;
a := [seq(A075084(n),n=1..120)]; # N. J. A. Sloane, Oct 20 2024

Mathematica

Table[n - PrimePi[2n] + PrimePi[n - 1] + 1, {n, 2, 75}]

PARI

a(n) = if (n>1, n - primepi(2*n) + primepi(n-1) + 1, 0); \\ Michel Marcus, Oct 21 2024

Python

from sympy import primepi
def A075084(n): return n+primepi(n-1)-primepi(n<<1)+1 if n>1 else 0 # Chai Wah Wu, Oct 20 2024

Formula

a(n) = n - pi(2n) + pi(n-1) + 1, for n>1.

Extensions

Edited by Robert G. Wilson v, Sep 12 2002

Definition clarified by N. J. A. Sloane, Oct 20 2024